Hierarchical modeling of high-resolution player-tracking data

Dan Cervone, NYU
June 14, 2016
ISBA World Meeting

Joint work with Luke Bornn, Alex D'Amour, Alex Franks, Kirk Goldsberry, Andrew Miller

1. Model the data generating process

   • Features of a DGP map to analytics of interest
   • DGP components can be interchanged and reused to answer new questions

2. Use qualitative and subjective knowledge

   • We're all domain experts when analyzing sports data

3. Hierarchical modeling helps

   • Efron: “enormous data sets often consist of enormous numbers of small sets of data”
   • Sports data have natural hierarchy and structured variation
   • Shrinkage and regularization necessary for useful inferences

Let \( \Omega \) be the space of all possible basketball possessions. For \( \omega \in \Omega \)

\[ \begin{align*} \nu_t &= \mathbb{E}[X | \mathcal{F}^{(Z)}_t] \\ &= \int_{\Omega} X(\omega) \mathbb{P}(d \omega | \mathcal{F}^{(Z)}_t) \end{align*} \]

Is this a regression-style prediction problem?

• Data are not traditional input/output pairs
• No guarantee of stochastic consistency

Stochastic process model for basketball:

• \( \mathbb{P}(Z_{t + \epsilon} | \mathcal{F}^{(Z)}_t) \) full-resolution transition kernel
• Allows Monte Carlo calculation of \( \nu_t \) by simulating future possession paths
• \( Z_t \) is high dimensional and includes discrete events (passes, shots, turnovers)

Let \( M(t) \) be the event that an event (pass/shot/turnover) occurs in \( (t, t + \epsilon] \)

Modeling movement and events are separate but complementary tasks:

\[ \begin{align*} \prod_{t} \mathbb{P}(Z_{t + \epsilon} | \mathcal{F}^{(Z)}_t) & = \Bigg( \prod_{t} \mathbb{P}(Z_{t + \epsilon}|M(t)^c, \mathcal{F}^{(Z)}_t)^{\mathbf{1}[M(t)^c]} \Bigg) & \hspace{0cm} \text{(player movement)} \\ & \times \Bigg( \prod_{t} \mathbb{P}(Z_{t + \epsilon}|M(t), \mathcal{F}^{(Z)}_t)^{\mathbf{1}[M(t)]} \Bigg) & \hspace{0cm} \text{(player reaction to events)} \\ & \hspace{-3cm} \times \Bigg( \prod_{t} \mathbb{P}(M(t)^c | \mathcal{F}^{(Z)}_t)^{\mathbf{1}[M(t)^c]} \mathbb{P}(M(t)|\mathcal{F}^{(Z)}_t)^{\mathbf{1}[M(t)]} \Bigg) & \hspace{0cm} \text{(event occurrences)} \end{align*} \]

Recall   \( M(t) \) denotes a pass, shot, or turnover in \( (t, t + \epsilon] \)

• Six different event types (4 pass options, shot, turnover), indexed by \( j \): \( \cup_{j=1}^6 M_j(t) = M(t) \)
• Event hazard:   \( \lambda_j(t) = \lim_{\epsilon \rightarrow 0} \frac{\mathbb{P}(M_j(t) | \mathcal{F}^{(Z)}_t)}{ \epsilon} \)

\[ \log(\lambda_j(t)) = [\mathbf{W}_j^{\ell}(t)]'\boldsymbol{\beta}_j^{\ell} + \xi_j^{\ell}\left(\mathbf{z}^{\ell}(t)\right) \]

• \( \ell \) indexes players
• \( \mathbf{W}_j^{\ell}(t), \boldsymbol{\beta}_j^{\ell} \): time-varying covariates and coefficients
• \( \xi_j^{\ell} \): spatial effect on log-hazard

Preprocessing

• Preprocessing involves low-resource, highly parallelizable tasks
• Parameter estimation involves several CPU- and memory-intensive tasks
• Calculating EPV from parameter estimates involves low-resource, highly parallelizable tasks

Assume for at-bat \( i \) for player \( j \):

• \( \mathbb{P}(Y_{ji} = 1 | X_{ji} = x) = p_j(x) \)
• Player's true batting average is \( \mu_j = \mathbb{E}[p_j(X)] \)

Traditional batting average is calculated using \( \hat{\mu}_j = \frac{1}{n_j}\sum_{i=1}^{n_j} Y_{ji} \)

\[ \begin{align*} \mathbb{E}[\hat{\mu}_j] &= \mu_j \\ \mathbb{Var}[\hat{\mu}_j] &= \frac{1}{n_j}(\mathbb{E}[p_j(X)(1 - p_j(X))] + \mathbb{Var}[p_j(X)]) \end{align*} \]

What if we knew \( p_j(X) \)?

• Batted ball trajectory is a sufficient statistic

Consider expected batting average \( \mu^E_j = \frac{1}{n_j}\sum_{i=1}^{n_j} p_j(x_i) \)

Outstanding challenges in modeling tracking data

• Measurement error
• Data size
• Heterogeneous data sources

Thanks to:

• Luke Bornn, Alex D'Amour, Alex Franks, Kirk Goldsberry, Andrew Miller.
• Moore/Sloan Foundations.

Traditional batting average: